I think the only way the coup succeeds is if Bill Paxon allows Dick Armey to stay in line for the Speakership, taking Majority Leader or Whip with Tom DeLay moving up and Dennis Hassert sidelined as a Gingrich loyalist.
Paxon and his wife Susan Molinari stay in Congress, Paxon has a future as a potential Speaker of the House. Molinari, rising star of the 1996 RNC, is not sidelined but if she stays in politics her future seems to be outside of Congress if Paxon remains there. She has been rumored for Senate or NYC mayor IOTL, but given the shift in politics she has a limited window for those runs. Maybe NYC Mayor in 2001? Or alternatively, she would be a visible running mate choice, Geraldine Ferraro comparisons aside.
Gingrich if he plays it right can step away from Congress and avoid connections with an impeachment push, being less toxic for a 2000 or 2004 Presidential run.
House GOP was really the inmates running the asylum by then, so even without Gingrich at the helm there is still probably impeachment fever the following year, assuming no butterflies with the Starr probe.